Minors
and Cofactors: Introduction; Expanding Along a Row (page
1 of 3)

Finding the determinant
of a 2×2
matrix is easy: You just do the criss-cross multiplication, and subtract:

The process for 3×3
matrices, while a bit messier, is still pretty straightforward: You add
repeats of the first and second columns to the end of the determinant,
multiply along all the diagonals, and add and subtract according to the
rule:

But for 4×4's
and bigger determinants, you have to drop back down to the smaller 2×2
and 3×3
determinants by using things called "minors" and "cofactors".

A "minor" is
the determinant of the square matrix formed by deleting one row and one
column from some larger square matrix. Since there are lots of rows and
columns in the original matrix, you can make lots of minors from it. These
minors are labelled according to the row and column you deleted. So if
you were to go, say, to the a2,4
entry from some matrix
A
and cross out the row and column that pass through that entry (that is,
if you remove the second
row and the fourth column of the matrix),
the determinant of the new (and smaller) matrix is called "the minor
M2,4".

the matrix A

cross out all
entries sharing arow or column with entry a2,4

the minor M2,4

Once you find a minor Mi,
j, you take
the subscript on the name of the minor (the "i,
j" part) and
add the two numbers i
and j.
Whatever result you get from this addition, make this value the power
on –1,
so you get "+1"
or "–1",
depending on whether i
+ j is even or
odd. Then multiply this on the minorMi,
j. This gives
you the "cofactor" Ai,
j. That is:

Okay, yeah; that probably
didn't make much sense. Here's
another way of saying it:

You've got some matrix
A.
You need to find the determinant of it.

It's too big to find by
the simpler methods, so you'll have to find it by "expanding along
a row or column".

Your first step in this
"expanding" will be picking a row or a column. Let's say you
pick the third row.

For each entry in the
third row, you will find the cofactor of that entry and multiply the
entry by its cofactor. That is, for the a3,1
entry of A,
you will find the cofactor A3,1,
and then you'll multiply the cofactor by the a3,1
entry: (a3,1)(A3,1).
For entry a3,2,
you will find the cofactor A3,2,
and multiply: (a3,2)(A3,2).
And so forth.

Then you will add up all
of these products: (a3,1)(A3,1)
+ (a3,2)(A3,2) + (a3,3)(A3,3)
+ ....

The resulting sum is the
value of the determinant of the matrix A.

(The above mess is why nobody
does determinants by hand if it can be avoided: there's just so
much error-prone mindless grunt-work involved.)

Weird fact: It doesn't
matter which row or column you use for your expansion; you'll get the
same value regardless. But this flexibility can be useful.

Find the determinant
of the following matrix by expanding (a) along the first row and (b)
along the third column. (c) Compare the results of each expansion.

(a) To expand along the
first row, I need to find the minors and then the cofactors of the first-row
entries: a1,1,
a1,2, a1,3,
and a1,4.